1.4. STRAINED EPITAXY: STRAIN TENSOR 21
to generate dislocations. In simplistic theories this occurs at an overlayer thickness called the
critical thickness ,dc, which is approximately given by
dc∼=aS
2 ||(1.3.2)
whereaSis the lattice constant of the substrate andthe lattice mismatch. In reality, the point in
growth where dislocations are generated is not so clear cut and depends upon growth conditions,
surface conditions, dislocation kinetics, etc. However, one may use the criteria given by equation
1.3.2 for approximately characterizing two regions of overlayer thickness for a given lattice
mismatch. Below critical thickness, the overlayer grows without dislocations and the film is
under strain. Under ideal conditions above critical thickness, the film has a dislocation array,
and after the dislocation arrays are generated, the overlayer grows without strain with its free
lattice constant.
If the strain value is greater than 0.03 one can still have strained epitaxy but the growth occurs
in the island mode where islands of the over-layer are formed. Such self-assembled islands are
being used for quantum dot structures.
Epitaxy beyond the critical thickness is important to provide new effective substrates for new
material growth. For these applications the key issues center around ensuring that the disloca-
tions generated stay near the overlayer-substrate interface and do not propagate into the overlayer
as shown in figure 1.17. A great deal of work has been done to study this problem. Often thin
superlattices in which the individual layers have alternate signs of strain are grown to “trap” or
“bend” the dislocations. It is also useful to build the strain up gradually.
In recent years, the GaN material system has seen much progress in electronic and optoelec-
tronic applications. Since GaN substrates are still not readily available, it is typically grown on
Al 2 O 3 (sapphire) or SiC , neither of which are closely lattice matched to GaN. The resulting
material is therefore highly dislocated. Many of the dislocations propagate upwards and are
terminated at the surface. In figure 1.18a, we show a cross-sectional transmission electron mi-
croscope image of GaN grown on sapphire. The vertical lines propagating upwards from the
highly defective interface are dislocations. Figure 1.18b is an atomic force microscope (AFM)
image of the GaN surface. The black pits are dislocations that have propagated upwards. Also
evident are the atomic steps that are typical of GaN surfaces. Such surface reconstructions were
described in section 1.2.5. Note that these atomic steps are always terminated by a dislocation.
In figure 1.18c, we show an AFM image of the surface of dislocation-free GaN. In contrast to
the dislocated material in figure 1.18b, there are no pits visible on the surface, and the surface
step structure is smooth and continuous.
1.4 STRAINEDEPITAXY:STRAINTENSOR
Growth of an epitaxial layer whose lattice constant is close, but not equal, to the lattice con-
stant of the substrate can result in a coherent strain. What is the strain tensor in such epitaxy?
The strain tensor determines how the electronic properties are altered. If the strain is small one
can have monolayer-by-monolayer. In this case the lattice constant of the epitaxial layer in the
directions parallel to the interface is forced to be equal to the lattice constant of the substrate.